Hausdorff dimension for piecewise monotonic maps. (English) Zbl 0687.58013

Author’s abstract: “In this paper certain closed T-invariant subsets R of [0,1], where T is an expanding piecewise monotonic map on the interval, are considered. It is shown that the Hausdorff dimension of R is equal to the unique zero of \(t\mapsto p(R,T,-t \log | T'|)\), where p denotes the pressure.”
The result holds in particular for closed sets R satisfying \(T^{- 1}R=R\), for sets \(R=\cap^{\infty}_{j=0}\overline{X\setminus T^{- j}G}\) where TG\(\subseteq G\), for R being a topologically transitive component of the center of (X,T) [F. Hofbauer, Probab. Theory Relat. Fields 72, 359-386 (1986; Zbl 0578.60069)], and for certain sets investigated by M. Urbański [Ergodic Theory Dyn. Syst. 7, 627-645 (1987; Zbl 0653.58031)]. All results are proved in a more general setting: The role of Lebesgue measure on [0,1] (i.e. the role of euclidean distance) for the definition of Hausdorff dimension can be played by a general Borel measure m on [0,1] with full topological support.
Reviewer: G.Keller


37B99 Topological dynamics
37A99 Ergodic theory
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
28D99 Measure-theoretic ergodic theory
54H20 Topological dynamics (MSC2010)
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